Caribou and reindeer are the same species. Eurasian and American caribou were cut off by rising sea levels, same as wolves and humans, and at the same time. We don't consider Indians and Europeans two different species, now do we?
You're quite right. But they're different sub-species of reindeer.
Socially the North American Reindeer are less friendly and more migratory then European ones. And European reindeer tend to be stockier and shorter with smaller horns. Its like comparing a North American mountain goat and an Asian Wild Goat, essentially the same physically but the North American mountain goat will kill you.
And I will call wild North American reindeer Caribou to help differentiate domestic reindeer from wild reindeer. Its a bit easier for me, and helps avoid confusion.
The big question is how large a range North American reindeer would have: what climates can they live in and how quickly they would spread (I think they could spread a lot faster than planet crops). Like llamas, they can be used to carry stuff but are too small to ride. It would be interesting if somebody invented the sledge, making winter travel easier (no wheels though

).
Actually
reindeer are ridden in Mongolia and throughout Siberia. Not sure how common reindeer riding is in Europe though. And if the Indians have a good reason to invent a wheel I think they would. Having an animal that can pull a reasonably large load across the prairies could be the key.
As for how far they can spread. Since caribou live in Yellowstone park and can be found in some places just south of Lake Superior (northernmost great lake) with human help they could probably live down to about New York with no real physical changes. I'm not sure how easily it would be to breed a more heat resistant reindeer but if North America were to herd reindeer for 3 or 4 thousand years I'm sure at least one breed could become heat resistant. With the tribes and civilizations to the south, there would be a reason to try.
The fun thing with the inuit is that the enormous amount of founder effect (the lapps still use them so I suspect they still have some worth) their many migrations had can lead to similarly enormous butterflies: when they landed in the Americas, they knew metallurgy, they knew how to make compound bows and they likely had reindeers domesticated, but since it was migrations of relatively small tribes, if your specialists die without an apprentice, you're going to be screwed especially if you can't spare some of your subsistance people in the band. There's a phase where what bonework survives even shows signs of struggling, with the really elaborate bonework being either very early or restarting around the Dorset era; it's altogether possible that they were caught completely unprepared by the little ice age leading to stuff like that.
Besides, Inuit reindeer archers has a slight feeling that it could be awesome - but if it had survived I suspect they would have displaced a lot of the people to the south of them, and maybe formed "yet another west coast mini-civilization" - if they hadn't displaced them all further to the south...
I didn't know that the bone carving had declined at one point, I'd always assumed it had continually improved. Cool.
I don't think they'd be a west coast civilization. Too little land and too easy to fish there. Having a reindeer civilization along the edge of the Rockies and following the rivers in the prairies seems like a more plausible idea.
And the Inuit reindeer archers are awesome. Unfortunately reindeer don't seem to be set up for mounted combat, so there couldn't be any reindeer knights.
Reindeer are relative poor food, and both as draft and riding (you can ride reindeer) they're restricted by size and climate. Don't get me wrong this will significant increase the population density of Native American Canada to the levels of Siberia and Lappland. Reindeer seem to have been domesticated quite early, but even with that it didn't really make inroad south, even at times where people farther south lacked access to draft animals.
I disagree.
They're not as good as horses or cows, but they would be a vast step up from what the Indians had in OTL. Being able to carry more goods across long distances, having milk and meat without hunting for it, manure for fields, a possibility of plowing even if its in lighter soil than a horse or cow can manage, would be quite useful.
The problem with reindeer in Europe is as you said reindeer are middling animals. Not as good as a horse or cow, only slightly better than a pig (reindeer have more variety, pigs have better and more meat), and goats and sheep offer wool along with everything else. Why would they choose reindeer that are new and not as good as what they had?
Now if the reindeer were a recent domesticate in North America, there wouldn't be enough time to breed a southern variety, and you'd be right. But if the Indians had time to make some different breeds and varieties then the Southern Indians could use them. The Reindeer would probably never cross a desert, but a breed could possibly be made to live in the swamps along the Mississippi.
It would effect European colonization of North America. If the Great Lakes, St. Lawrence River region get a denser agriculutral population, perhaps even as far south as New England, early European settlement is going to be very different.
In the west coast the Willamette Valley could host a much larger settled population. Considering this region was untouched for quite awhile, conceivably the local tribes could absorb enough modern technology to build a state of their own, like the Cherokees but no Trail of Tears.
Thats what I'm thinking. While the Indians wouldn't be technologically equal to Europe, the larger population base and better agriculture would change things, especially in the north.